We studied the dynamics of dissolved water in aluminosilicate glasses with the compositions NaAlSi3O8⋅0.3H2O, NaAlSi3O8⋅1.3H2O and Ca0.5AlSi3O8⋅1.3H2O using quasielastic neutron scattering. As shown by near-infrared spectroscopy on these samples, H2O molecules are the predominant hydrous species in the water-rich glasses whereas OH groups bound to tetrahedrally coordinated cations are predominant at low water contents. Backscattering and time-of-flight methods were combined to investigate motional correlation times in the range between 0.2ps and 2ns. For the water-rich glasses an elastic scan between 2K and 420K shows that the dynamical processes set in at lower temperatures in the Ca-bearing glass than in the Na-bearing glass. This is corroborated by the broadening of the inelastic spectra S(Q,ω). The shape of the scattering function S(Q,t) suggests a distribution of activation barriers for the motion of hydrous species in the disordered structure of the glass. The distribution is narrower and the average activation energy is smaller in the Ca-bearing glass than in the Na-bearing glass. No indication for dynamics of hydrous species was found at temperatures up to 520K in the water-poor glass NaAlSi3O8⋅0.3H2O containing dissolved water mainly in the form of OH groups. It is concluded that H2O molecules are the dynamic species in the above-mentioned time regime in the water-rich glasses. The dynamic process is probably a rotation of H2O molecules around their bisector axis. © 2005 The American Physical Society.